Algebraic algorithms for approximation of special functions

Abstract:
Problem. To construct algorithms for computing polynomials (approximation of special functions (SFs)) with a computer algebra system (CAS).

Relevance. CAS approximates SFs with the Fourier-Chebyshev series and computes their values. The approximation accuracy is limited by the accuracy of computing values of SFs.

Method. We propose algebraic algorithms within the Lanczos tau-method, the Dzyadyk a-method etc. including algorithms allowing for symmetry.

Result. We construct algorithms and algebraic programming system (APS) procedures which solve all types of equations defining SFs. The optimality of found polynomials by accuracy of SF approximation is proved.

Symmetry. Procedures involving symmetries compute polynomials of orders twice greater than procedures which do not use symmetries.